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# lect_39 - Graph Coloring Margaret M Fleck 3 May 2010 This...

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Graph Coloring Margaret M. Fleck 3 May 2010 This lecture discusses the graph coloring problem (section 9.8 of Rosen). 1 Announcements Makeup quiz last day of classes (at the start of class). Your room for the final exam (Friday the 7th, 7-10pm) is based on the first letter of your last name: A-H Roger Adams Lab 116 I-W Business Instructional Facility (515 E. Gregory opposite Armory) 1001 X-Z Roger Adams Lab 116 The conflict exam (Monday the 10th, 1:30-4:30pm) is in 269 Everett Lab. 2 The problem We’re only going to talk about simple undirected graphs. Remember that two vertices are adjacent if they are directly connected by an edge. 1

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A coloring of a graph G assigns a color to each vertex of G , with the restriction that two adjacent vertices never have the same color. The chro- matic number of G , written χ ( G ), is the smallest number of colors needed to color G . For example, only three colors are required for this graph: R G G B R But K 4 requires 4: R G B Y 3 Computing colorings For certain classes of graphs, we can easily compute the chromatic number. For example, the chromatic number of K n is n , for any n . Notice that we have to argue two separate things to establish that this is its chromatic number: K n can be colored with n colors. K n cannot be colored with less than n colors For K n , both of these facts are fairly obvious. Assigning a different color to each vertex will always result in a well-formed coloring (though it may be 2
a waste of colors). Since each vertex in K n is adjacent to every other vertex, no two can share a color. So fewer than n colors can’t possibly work. Similar, the chromatic number for K n,m is 2. We color one side of the graph with one color and the other side with a second color. In general, however, coloring requires exponential time. There’s a couple specific versions of the theoretical problem. I could give you a graph and ask you for its chromatic number. Or I could give you a graph and an integer k and ask whether k colors is enough to be able to color G . These problems are all “NP-complete” or “NP-hard.” That is, they apparently require exponential time to compute, even though in some cases it’s easy to check that their output is correct. (They are exponential if the “ P ” and NP ” classes of algorithms are actually different, which is one of the big outstanding problems in computer science.)

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lect_39 - Graph Coloring Margaret M Fleck 3 May 2010 This...

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